Problems

A set includes weights weighing $1$ gram, $2$ grams, $4$ grams, ... (all powers of the number $2$), and in the set some of the weights might be the same. Weights were placed on two cups of the scales so that the scales are in balance. It is known that on the left cup, all weights are different. Prove that there are as many weights on the right cup as there are on the left.

Generally, when a line intersects a circle, it creates two different points of intersection. However, sometimes there is only one point. In such case we say the line is tangent to the circle. For example on the picture below the line $CD$ intersects the circle at two points $D$ and $E$ and the line $CB$ is tangent to the circle. To solve the problems today we will need the following theorem.
Theorem: The radius $AB$ is perpendicular to the tangent line $BC$.

Two lines $CD$ and $CB$ are tangent to a circle with the center $A$ and radius $R$, see the picture. The angle $\mathrm{\angle }BCD$ equals ${120}^{\circ }$. Find the length of $BD$ in terms of $R$.

Given two circles, one has centre $A$ and radius $r$, another has centre $C$ and radius $R$. Both circles are tangent to a line at the points $B$ and $D$ respectively and the angles $\mathrm{\angle }CED=\mathrm{\angle }AEB={30}^{\circ }$. Find the length of $AC$ in terms of $r$ and $R$.

Consider a triangle $CDE$. The lines $CD$, $DE$, and $CE$ are tangent to a circle with centre $A$ at the points $F,G$, and $B$ respectively. We also have that the angle $\mathrm{\angle }DCE={120}^{\circ }$. Prove that the length of the segment $AC$ equals the perimeter of the triangle $CDE$.

A circle with center $A$ is tangent to the lines $CB$ and $CD$, see picture. Find the angles of the triangle $BCD$ if $BD=BC$.

Take two circles with a common centre $A$. A chord $CD$ of the bigger circle is tangent to the smaller one at the point $B$. Prove that $B$ is the midpoint of $CD$.

Prove that the lines tangent to a circle in two opposite points of a diameter are parallel.

$CD$ is a chord of a circle with centre $A$. The line $CD$ is parallel to the tangent to the circle at the point $B$. Prove that the triangle $BCD$ is isosceles.

Four lines, intersecting at the point $D$, are tangent to two circles with a common center $A$ at the points $C,F$ and $B,E$. Prove that there exists a circle passing through all the points $A,B,C,D,E,F$.